Vector and Tensor Analysis with Applications

Cover......Page 1
Editor's Preface......Page 4
Contents......Page 6
1.1. Vectors and Scalars......Page 10
1.1.1. Free, sliding and bound vectors......Page 11
1.2.1. Addition of vectors......Page 12
1.2.2. Subtraction of vectors......Page 14
1.2.3. Projection of a vector onto an axis......Page 15
1.3.1. Linear dependence and linear independence of vectors......Page 16
1.3.2. Expansion of a vector with respect to other vectors......Page 17
1.3.3. Bases and basis vectors......Page 18
1.3.4. Direct and inverse transformations of basis vectors......Page 22
1.4.1. The scalar product......Page 23
1.4.2. The vector product......Page 25
1.4.3. Physical examples......Page 28
1.5.1. The scalar triple product......Page 29
1.5.2. The vector triple product......Page 30
1.6.1. Reciprocal bases......Page 32
1.6.2. The summation convention......Page 35
1.6.3. Covariant and contravariant components of a vector......Page 36
1.6.4. Physical components of a vector......Page 38
1.6.5. Relation between covariant and contravariant components......Page 40
1.6.6. The case of orthogonal bases......Page 42
1.7.1. Vector functions of a scalar argument......Page 44
1.7.2. The derivative of a vector function......Page 45
1.7.3. The integral of a vector function......Page 46
Solved Problems......Page 47
Exercises......Page 63
2.1. Preliminary Remarks......Page 68
2.2. Zeroth-Order Tensors (Scalars)......Page 69
2.3. First-Order Tensors (Vectors)......Page 70
2.3.1. Examples......Page 71
2.4. Second-Order Tensors......Page 72
2.4.1. Examples......Page 73
2.4.2. The stress tensor......Page 75
2.4.3. The moment of inertia tensor......Page 77
2.4.4. The deformation tensor......Page 79
2.4.5. The rate of deformation tensor......Page 81
2.5. Higher-Order Tensors......Page 85
2.6. Transformation of Tensors under Rotations about a Coordinate Axis......Page 86
2.7. Invariance of Tensor Equations......Page 90
2.8. Curvilinear Coordinates......Page 91
2.8.2. Coordinate curves......Page 93
2.8.3. Bases and coordinates axes......Page 94
2.8.4. Arc length. Metric coefficients......Page 95
2.9.1. Covariant, contravariant and mixed components of a tensor......Page 97
2.9.2. The tensor character of g_ik, g^ik and g_i^k......Page 98
2.9.4. Physical components of a tensor. The case of orthogonal bases......Page 99
2.9.5. Covariant, contravariant and mixed tensors as such......Page 100
Solved Problems......Page 103
Exercises......Page 109
3.1. Addition of Tensors......Page 112
3.3. Contraction of Tensors......Page 113
3.4.1. Symmetric and antisymmetric tensors......Page 114
3.4.2. Equivalence of an antisymmetric second-order tensor to an axial vector......Page 116
3.5.1. Statement of the problem......Page 118
3.5.2. The two-dimensional case......Page 119
3.5.3. The three-dimensional case......Page 122
3.5.4. The tensor ellipsoid......Page 127
3.6. Invariants of a Tensor......Page 130
3.7.1. Proper and improper transformations......Page 131
3.7.2. Definition of a pseudotensor......Page 133
3.7.3. The pseudotensor ε_jkl......Page 134
Solved Problems......Page 135
Exercises......Page 140
4.1.1. Tensor functions of a scalar argument......Page 143
4.1.3. Line integrals. Circulation......Page 144
4.2.1. Gauss' theorem......Page 146
4.2.2. Green's theorem......Page 148
4.2.3. Stokes' theorem......Page 150
4.2.4. Simply and multiply connected regions......Page 153
4.3.1. Level surfaces......Page 154
4.3.2. The gradient and the directional derivative......Page 155
4.3.3. Properties of the gradient. The operator V......Page 158
4.3.4. Another definition of grad φ......Page 159
4.4.1. Trajectories of a vector field......Page 160
4.4.2. Flux of a vector field......Page 161
4.4.3. Divergence of a vector field......Page 164
4.4.4. Physical examples......Page 166
4.4.5. Curl of a vector field......Page 170
4.4.6. Directional derivative of a vector field......Page 173
4.5. Second-Order Tensor Fields......Page 175
4.6. The Operator V and Related Differential Operators......Page 177
4.6.1. Differential operators in orthogonal curvilinear coordinates......Page 180
Solved Problems......Page 183
Exercises......Page 191
5.1.1. Covariant differentiation of vectors......Page 194
5.1.2. Christoffel symbols......Page 196
5.1.3. Covariant differentiation of tensors......Page 199
5.1.4. Ricci's theorem......Page 200
5.1.5. Differential operators in generalized coordinates......Page 201
5.2. Integral Theorems......Page 205
5.2.1. Theorems related to Gauss' theorem......Page 206
5.2.2. Theorems related to Stokes' theorem......Page 207
5.2.3. Green's formulas......Page 210
5.3.1. Equations of fluid motion......Page 212
5.3.2. The momentum theorem......Page 217
5.4. Potential and Irrotational Fields......Page 220
5.4.1. Multiple-valued potentials......Page 222
5.5. Solenoidal Fields......Page 225
5.6.1. Harmonic functions......Page 228
5.6.2. The Dirichlet and Neumann problems......Page 231
5.7. The Fundamental Theorem of Vector Analysis......Page 232
5.8.1. Maxwell's equations......Page 235
5.8.2. The scalar and vector potentials......Page 237
5.8.3. Energy of the electromagnetic field. Poynting's vector......Page 239
Solved Problems......Page 241
Exercises......Page 256
BIBLIOGRAPHY......Page 260
INDEX......Page 262